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We consider the nonlinear optimisation of irreversible mixing induced by aninitial finite amplitude perturbation of a statically stable density-stratifiedfluid. A constant pressure gradient is imposed in a plane two-dimensionalchannel. We consider flows with a finite P\'eclet number $Pe=500$ and Prandtlnumber $Pr=1$, and a range of bulk Richardson numbers $Ri_b \in [0,1]$. We usethe constrained variational direct-adjoint-looping (DAL) method to solve twooptimization problems, extending the optimal mixing results of Foures et al.(2014) to stratified flows, where the mixing of the scalar density has anenergetic cost, and thus has an observable dynamic effect. We identify initialperturbations of fixed finite kinetic energy which maximise the time-averagedkinetic energy developed by the perturbations over a finite time interval, andinitial perturbations that minimise the value of a `mix-norm', as defined byThiffeault (2012) and shown by Foures et al. (2014) to be a computationallyefficient and robust proxy for identifying perturbations that minimise thevariance of a scalar distribution at a target time. We demonstrate, for allbulk Richardson numbers considered, that thetime-averaged-kinetic-energy-maximising perturbations are significantlysuboptimal at mixing compared to the mix-norm-minimising perturbations. Byconsidering the time evolution of the kinetic energy and potential energyreservoirs, we find that mix-norm-minimising optimal perturbations lead to aflow which, through Taylor dispersion, very effectively converts perturbationkinetic energy into `available potential energy', which in turn leads rapidlyand irreversibly to thorough and efficient mixing, with little energy returnedto the kinetic energy reservoirs.

Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite P\'eclet and Richardson numbers